Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.
Theorem selected for verification:OAI.CHObstruction.main
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Definitions needed to read it
Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.
structureIsSubstructure (M N : RelModel.{u} L) : Propwhere subset : M.carrier ⊆ N.carrier restrict : ∀ (n) (r : L n) (a : Fin n → M.carrier), M.rel n r a ↔ N.rel n r (fun i => ⟨(a i).val, subset (a i).property⟩)
structureIso (M N : RelModel.{u} L) where toEquiv : M.carrier ≃ N.carrier map_rel : ∀ (n) (r : L n) (a : Fin n → M.carrier), M.rel n r a ↔ N.rel n r (toEquiv ∘ a)
defLSBound (θ : Cardinal.{u}) : Prop := ℵ₀ ≤ θ ∧ Cardinal.lift.{u, 0} (# (Σ n, L n)) ≤ θ ∧ ∀ M, K.objects M → ∀ A : ZFSet.{u}, A ⊆ M.carrier → ∃ N, K.objects N ∧ K.strong N M ∧ A ⊆ N.carrier ∧ N.carrier.card ≤ A.card + θ
strong_objects : ∀ {M N}, K.strong M N → K.objects M ∧ K.objects N refl : ∀ {M}, K.objects M → K.strong M M antisymm : ∀ {M N}, K.strong M N → K.strong N M → M = N trans : ∀ {M N P}, K.strong M N → K.strong N P → K.strong M P substructure : ∀ {M N}, K.strong M N → M.IsSubstructure N iso_objects : ∀ {M N}, K.objects M → Nonempty (M.Iso N) → K.objects N iso_strong : ∀ {M N M' N'} (h : K.strong M N) (e : M.Iso M') (f : N.Iso N'), (∀ x : M.carrier, (e.toEquiv x).val = (f.toEquiv ⟨x.val, (substructure h).subset x.property⟩).val) → K.strong M' N' coherence : ∀ {M₀ M₁ M₂}, M₀.IsSubstructure M₁ → K.strong M₀ M₂ → K.strong M₁ M₂ → K.strong M₀ M₁ chain_union : ∀ (γ : Ordinal.{u}), γ ≠ 0 → ∀ (F : γ.ToType → RelModel.{u} L), (∀ i, K.objects (F i)) → (∀ i j, i ≤ j → K.strong (F i) (F j)) → ∃ U, K.objects U ∧ RelModel.IsUnion F U ∧ (∀ i, K.strong (F i) U) ∧ (∀ N, K.objects N → (∀ i, K.strong (F i) N) → K.strong U N) downward : ∃ θ, K.LSBound θ
Atlas selected this possible match by comparing words in the Lean statement with every statement in the family. No review of whether the statements mean the same thing is recorded.
Words of the Lean statement, its names split into words, that also appear in the theorem: downward, beth, ls, countable, categorical, two, models and cardinal. Found in the paper’s title, which its main theorem states: ch and threshold. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: transfer, continuum, omega, elementary, nonisomorphic, relational, abstract, language and four more.
Statements in A CH obstruction to a prescribed categoricity threshold come first because the release lists only that paper for this Lean statement.
Theorem 1.1 of A CH obstruction to a prescribed categoricity threshold
Theorem
Assume the continuum hypothesis. There is an abstract elementary class K in a countable finitary relational language such that
(i)
\LS(K)=\aleph_0;
(ii)
K has at least two nonisomorphic models of cardinality \beth_{\omega_2}=H(K);
(iii)
putting \Lambda=\beth_{(2^{\aleph_1})^+}, the class K is categorical in every cardinal \mu\ge\Lambda.
In particular, \lambda=\Lambda^+>H(K) is a categoricity cardinal from which downward transfer to H(K) fails.
Assume the continuum hypothesis. There is an abstract elementary class
\(K\) in a countable finitary relational language such that
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(\LS(K)=\aleph_0\);
\item \(K\) has at least two nonisomorphic models of cardinality
\(\beth_{\omega_2}=H(K)\);
\item putting \(\Lambda=\beth_{(2^{\aleph_1})^+}\),
the class \(K\) is categorical in every cardinal
\(\mu\ge\Lambda\).
\end{enumerate}
In particular, \(\lambda=\Lambda^+>H(K)\) is a categoricity cardinal
from which downward transfer to \(H(K)\) fails.
Other possible paper matches, closest wording first
Theorem 3.13 of A CH obstruction to a prescribed categoricity threshold. Words of the Lean statement, its names split into words, that also appear in the theorem: substructures, nonempty, union, ls, aec, countable and four more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: elementary, abstract, relational, language and aleph.
Theorem 4.5 of A CH obstruction to a prescribed categoricity threshold. Words of the Lean statement, its names split into words, that also appear in the theorem: beth, countable, two and models. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: nonisomorphic and aleph.
Theorem 1.2 of Eventual categoricity for abstract elementary classes. Words of the Lean statement, its names split into words, that also appear in the theorem: ls, aec, threshold, categorical and cardinal. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: categoricity.
Proposition 7.2 of A CH obstruction to a prescribed categoricity threshold. Words of the Lean statement, its names split into words, that also appear in the theorem: tail, countable and model. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: categoricity.
Theorem 14.5 of Eventual categoricity for abstract elementary classes. Words of the Lean statement, its names split into words, that also appear in the theorem: tail, aec, categorical, model and cardinals. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: categoricity.
Theorem 1.3 of Eventual categoricity for abstract elementary classes. Words of the Lean statement, its names split into words, that also appear in the theorem: aec, tail, categorical and cardinals.
How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.
Reported Comparator runs
three independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet; Neal McBurnett’s openai-math-lean-tooling. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.
For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…
AcceptedTheir term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
AcceptedTheir term: “comparator-sandboxed-pass”.
Re-run through Comparator in a sandboxed virtual machine: accepted.
Each challenge’s solution is built and checked by Comparator at the pinned release, on Lean leanprover/lean4:v4.34.1, recording the verdict, the check time and the peak memory; the run was still in progress at this snapshot (192 accepted, 224 pending).
AcceptedTheir term: “accepted”.
Re-run through Comparator with Lean’s kernel: accepted (the check took 51 s; peak memory 6.1 GiB).
If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.
This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.
Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.
Do family 240’s Lean statements express the advertised result?
These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.
Neither group finds that the Lean statements express or imply the whole advertised result. They describe what the statements express in different terms.
Finds part of it stated, or a narrower or weaker statementTheir term: “partial”.
Only a special case is stated, or a co-equal headline claim has no challenge statement. Their record
Their note: PRIMARY CLAIM NOT STATED (only a secondary result is formalized). CHObstruction.main (hCH : CH) : ∃ L countable relational, ∃ K, K.IsAEC ∧ K.HasLSNumber ℵ₀ ∧ endpoint = hanf ℵ₀ ∧ K.TwoModels endpoint ∧ ∀ μ, tailThreshold ≤ μ → K.Categorical μ is the summary's last sentence (under CH a prescribed Hanf threshold need not suffice), formalized with hand-written AEC axioms on ZFSet models. The headline, Shelah's eventual categoricity conjecture proved in ZFC, is not stated (docs: 'The later canonical-point obstruction and the separate eventual-categoricity theorem are not included').
Definitions the group asks readers to compare with the paper: Hand-rolled abstract elementary classes: structure ClassData (objects, strong), IsAEC (iso-closure, partial order, coherence, chain unions over arbitrary γ.ToType, LS bound), LSBound/HasLSNumber, Categorical μ := (∃ M, objects M ∧ card = μ) ∧ ∀ M N of size μ, Nonempty (M.Iso N), hanf κ := beth (succ (2^κ)).ord; standard Shelah axioms but entirely custom, and antisymm is literal equality M = N of RelModels.
Finds part of it stated, or a narrower or weaker statementTheir term: “NARROWER — DECLARED”.
The challenge statements state less than the headline, and OpenAI’s Lean note says so. Their record
Their note: The headline's main claim — Shelah's eventual categoricity conjecture proved in ZFC (uniform threshold, transfer of categoricity through the tail) — is not stated; only the CH obstruction to the prescribed Hanf threshold is.
OpenAI’s own note on what this Lean statement covers
“Under the continuum hypothesis, the formalized result refutes the proposed transfer of categoricity down to the Hanf threshold. It gives an abstract elementary class in a countable relational language, with Löwenheim–Skolem number $\aleph_0$, that has two nonisomorphic models at $\beth_{\omega_2}=h(\aleph_0)$ but is categorical in every cardinal at least $\beth_{(2^{\aleph_1})^+}$.”
An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.
Supporting material
The complete Lean statement file
107 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.
Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.
Links in the file: 17 to its own declarations, 28 to Mathlib’s documentation. 15 declared names also link to the same declaration in OpenAI’s library. 7 identifiers stay plain because they resolve to no single declaration: variables Lean binds automatically, notation, or names the scan of the pinned sources did not find. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.
import Mathlibnamespace OAIuniverse unoncomputablesectionnamespace CHObstructionopen Cardinal Ordinal OrderstructureRelModel (L : ℕ → Type) wherecarrier : ZFSet.{u}rel : (n : ℕ) → L n → (Fin n → carrier) → Propnamespace RelModelvariable {L : ℕ → Type}structureIsSubstructure (M N : RelModel.{u} L) : Propwheresubset : M.carrier ⊆ N.carrierrestrict : ∀ (n) (r : L n) (a : Fin n → M.carrier), M.rel n r a ↔ N.rel n r (fun i => ⟨(a i).val, subset (a i).property⟩)structureIso (M N : RelModel.{u} L) wheretoEquiv : M.carrier ≃ N.carriermap_rel : ∀ (n) (r : L n) (a : Fin n → M.carrier), M.rel n r a ↔ N.rel n r (toEquiv ∘ a)defIsUnion {ι : Type u} (F : ι → RelModel.{u} L) (U : RelModel.{u} L) : Prop := (∀ i, IsSubstructure (F i) U) ∧ ∀ x : U.carrier, ∃ i, x.val ∈ (F i).carrierend RelModelstructureClassData (L : ℕ → Type) whereobjects : RelModel.{u} L → Propstrong : RelModel.{u} L → RelModel.{u} L → Propnamespace ClassDatavariable {L : ℕ → Type} (K : ClassData.{u} L)defLSBound (θ : Cardinal.{u}) : Prop := ℵ₀ ≤ θ ∧ Cardinal.lift.{u, 0} (# (Σ n, L n)) ≤ θ ∧ ∀ M, K.objects M → ∀ A : ZFSet.{u}, A ⊆ M.carrier → ∃ N, K.objects N ∧ K.strong N M ∧ A ⊆ N.carrier ∧ N.carrier.card ≤ A.card + θdefHasLSNumber (θ : Cardinal.{u}) : Prop := K.LSBound θ ∧ ∀ θ', K.LSBound θ' → θ ≤ θ'structureIsAEC : Propwherestrong_objects : ∀ {M N}, K.strong M N → K.objects M ∧ K.objects Nrefl : ∀ {M}, K.objects M → K.strong M Mantisymm : ∀ {M N}, K.strong M N → K.strong N M → M = Ntrans : ∀ {M N P}, K.strong M N → K.strong N P → K.strong M Psubstructure : ∀ {M N}, K.strong M N → M.IsSubstructure Niso_objects : ∀ {M N}, K.objects M → Nonempty (M.Iso N) → K.objects Niso_strong : ∀ {M N M' N'} (h : K.strong M N) (e : M.Iso M') (f : N.Iso N'), (∀ x : M.carrier, (e.toEquiv x).val = (f.toEquiv ⟨x.val, (substructure h).subset x.property⟩).val) → K.strong M' N'coherence : ∀ {M₀ M₁ M₂}, M₀.IsSubstructure M₁ → K.strong M₀ M₂ → K.strong M₁ M₂ → K.strong M₀ M₁chain_union : ∀ (γ : Ordinal.{u}), γ ≠ 0 → ∀ (F : γ.ToType → RelModel.{u} L), (∀ i, K.objects (F i)) → (∀ i j, i ≤ j → K.strong (F i) (F j)) → ∃ U, K.objects U ∧ RelModel.IsUnion F U ∧ (∀ i, K.strong (F i) U) ∧ (∀ N, K.objects N → (∀ i, K.strong (F i) N) → K.strong U N)downward : ∃ θ, K.LSBound θdefCategorical (μ : Cardinal.{u}) : Prop := (∃ M, K.objects M ∧ M.carrier.card = μ) ∧ ∀ M N, K.objects M → K.objects N → M.carrier.card = μ → N.carrier.card = μ → Nonempty (M.Iso N)defTwoModels (μ : Cardinal.{u}) : Prop := ∃ M N, K.objects M ∧ K.objects N ∧ M.carrier.card = μ ∧ N.carrier.card = μ ∧ ¬ Nonempty (M.Iso N)end ClassDatadefCH : Prop := (2 : Cardinal.{u}) ^ ℵ₀ = ℵ_ 1defhanf (κ : Cardinal.{u}) : Cardinal.{u} := beth (succ ((2 : Cardinal.{u}) ^ κ)).orddefendpoint : Cardinal.{u} := beth (ω_ 2)deftailThreshold : Cardinal.{u} := beth (succ ((2 : Cardinal.{u}) ^ (ℵ_ 1))).ordtheoremmain (hCH : CH.{u}) : ∃ (L : ℕ → Type), Countable (Σ n, L n) ∧ ∃ K : ClassData.{u} L, K.IsAEC ∧ K.HasLSNumber ℵ₀ ∧endpoint = hanf (ℵ₀ : Cardinal.{u}) ∧ K.TwoModels endpoint ∧ ∀ μ, tailThreshold ≤ μ → K.Categorical μ := bysorryend CHObstructionendend OAI
Every extracted statement of the accompanying papers
The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.
Assume the continuum hypothesis. There is an abstract elementary class
\(K\) in a countable finitary relational language such that
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(\LS(K)=\aleph_0\);
\item \(K\) has at least two nonisomorphic models of cardinality
\(\beth_{\omega_2}=H(K)\);
\item putting \(\Lambda=\beth_{(2^{\aleph_1})^+}\),
the class \(K\) is categorical in every cardinal
\(\mu\ge\Lambda\).
\end{enumerate}
In particular, \(\lambda=\Lambda^+>H(K)\) is a categoricity cardinal
from which downward transfer to \(H(K)\) fails.
A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.
Recorded comparisons with the paper
Atlas records no contributed comparison of this statement with the paper.
Contributed Comparator run reports
Atlas records no contributed report of running Comparator for this statement file.
Questions to resolve when comparing this statement with the paper
Both independent readers find this family's challenges state less than its headline, while the atlas's AI draft classes this file as stating a paper's theorem as the paper does; check which part of the headline it covers.